- September 19, 2023
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The powers of i is always equal to either one of these 4 numbers: 1, i , -1,-i. Here we will see why.
i0 = 1, this is because for any non-zero number, any number with exponent 0 is equal to 1.
i1 = i, this is because any number with exponent 1 is equal to itself, in this case, i itself.
i2 = -1, this can be explained by the definition of imaginary number, which says i is defined as
i3 = -i, since i2=-1, i3 = i2 * i = -1 * i = -i
i4 = 1 becausei4 = i3 * i = -i * i = – * = 1
We see that i4 = i0, so this is a cycle, so by prediction i5 equals to i, i6 equals -1, and so on…. and in fact that is true.
So i with the exponent n (any whole number from 0 to above) can be determined by the following rules:
Divide the exponent by 4 and find the remainder,
if the remainder is 0 (no remainder), then in equals 1;
if the remainder is 1, then in equals i;
if the remainder is 2, then in equals -1;
if the remainder is 3, then in equals -i.